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Chapter 4: Problem 24

\(y=3 \csc \left(1-\frac{x}{3}\right), \quad x=1, \quad d x=0.1\)

Short Answer

Expert verified
The final calculation of the function is the product of 3 and the reciprocal of the sine of 2/3.

Step by step solution

01

Understand the Function

Firstly, the function provided, \(y = 3 \csc \left(1-\frac{x}{3}\right)\) uses the trigonometric function cosecant, which is the reciprocal of the sine function.
02

Substitute the values for x and dx

Substitute \(x = 1\) and \(dx = 0.1\) into the function. Therefore, it becomes: \(y = 3 \csc \left(1-\frac{1}{3}\right)\). Simplify the fraction inside the bracket so that it becomes: \(y = 3 \csc \left(\frac{2}{3}\right)\).
03

Calculate the cosecant of 2/3

Now, calculate the cosecant of \(\frac{2}{3}\). As mentioned before, cosecant is the reciprocal of the sine function. So, we need to calculate \(\csc \left(\frac{2}{3}\right) = 1 / \sin \left(\frac{2}{3}\right)\).
04

Final Calculation

Now, multiply 3 with the obtained value of \(\csc \left(\frac{2}{3}\right)\) to get the final value of y.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosecant Function
The cosecant function, denoted as \(\csc \theta\), is one of the lesser-known trigonometric functions, primarily used in contexts where reciprocal identities are needed. It is defined as the reciprocal of the sine function. This means that for an angle \(\theta\), the cosecant is given by \(\csc \theta = \frac{1}{\sin \theta}\).
  • This definition implies that the cosecant function is only defined for values of \(\theta\) where the sine is not zero.
  • Cosecant is periodic like the sine function but has its own unique graph and properties.
The understanding of this function is crucial when working through problems involving reciprocal identities, especially in scenarios involving angle transformations or certain identities.
Sine Function
The sine function is a fundamental trigonometric function, commonly denoted as \(\sin \theta\). It measures the ratio of the opposite side to the hypotenuse in a right-angled triangle. This geometric interpretation is useful as it sets the stage for understanding wave patterns and harmonic motions.
  • A crucial property of the sine function is its periodicity, with a period of \(2\pi\), meaning that \(\sin(\theta + 2\pi) = \sin \theta\).
  • Sine values range between -1 and 1, and locations on the unit circle help determine these values.
In terms of the provided exercise, the sine function becomes necessary when calculating reciprocal trigonometric expressions, such as when finding the cosecant of a given angle.
Reciprocal Identities
Reciprocal identities are a set of important relationships in trigonometry that express trigonometric functions as the reciprocals of other functions. These identities include:
  • \(\csc \theta = \frac{1}{\sin \theta}\)
  • \(\sec \theta = \frac{1}{\cos \theta}\)
  • \(\cot \theta = \frac{1}{\tan \theta}\)
Understanding these identities is crucial for simplifying trigonometric expressions and solving equations. In the given exercise, knowing that the cosecant is the reciprocal of sine allows us to reframe and solve for the value of \(y\) by first calculating \(\sin \left(\frac{2}{3}\right)\) and then finding its reciprocal. This concept can drastically simplify many trigonometric computations and is a staple in calculus and other advanced mathematics areas.

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Most popular questions from this chapter

True or False If \(f^{\prime}(c)=0\) and \(f(c)\) is not a local maximum, then \(f(c)\) is a local minimum. Justify your answer.

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